Issue 
A&A
Volume 494, Number 2, February I 2009



Page(s)  623  636  
Section  Interstellar and circumstellar matter  
DOI  https://doi.org/10.1051/00046361:200810587  
Published online  20 November 2008 
Online Material
Appendix A: The reaction rate table used in the Nahoon modified chemical model
Table A.1: Reactions used in the model. The rate coefficients are given for 7 K. Reaction rates less than 10^{15} cm^{3} s^{1} are not taken into account in our models. Reference (1) corresponds to Gerlich (1990), references (2)(4) correspond to Hugo, OSU 07, and this paper respectively. For OSU 07, branching ratios involving spin states have been infered from quantum mechanical rules. For reactions involving grains, a grain radius of 0.1 m and a sticking coefficient of 1 have been considered. (5) Datz et al. (1995b) (6) Datz et al. (1995a), (7) Zhaunerchyk et al. (2008), (8) Larsson et al. (1997), (9) Molek et al. (2007).
Appendix B: On the rate coefficients for dissociation recombination of H_{3}^{+} and its isotopologues
The dissociative recombination (DR) rate coefficients for ortho and paraH_{3}^{+} have been published by Fonseca dos Santos et al. (2007). Here, we present the results obtained for all four H_{3}^{+} isotopologues. The DR rate coefficients for different species of the nuclear spin are calculated using the approach described in a series of papers devoted to DR theory for triatomic molecular ions. See Fonseca dos Santos et al. (2007); Kokoouline & Greene (2003a,b) for H_{3}^{+} and D_{3}^{+} calculations and Kokoouline & Greene (2004,2005) for H_{2}D^{+} and D_{2}H^{+}. The scope of this paper does not allow us to review the theoretical approach in detail. We only list its main ingredients.
The theoretical approach is fully quantum mechanical and incorporates no adjustable parameters. It relies on ab initio calculations of potential surfaces for the ground electronic state of the H_{3}^{+} ion and several excited states of the neutral molecule H_{3}, performed by Mistrik et al. (2001).
The total wave function of the system is constructed by an appropriate symmetrization of products of vibrational, rotational, electronic, and nuclear spin factors. Therefore, rovibronic and nuclear spin degrees of freedom are explicitly taken into account.
The electronic BornOppenheimer potentials for the four H_{3}^{+} (and H_{3}) isotopologues have the C_{3v} symmetry group. The C_{3v} symmetry group has a twodimensional irreducible representation E. The ion has a closed electronic shell. The lowest electronic state of the outer electron in H_{3} has pwave character. The pwave state of the electron also belongs to the E representation. Due to the JahnTeller theorem (Landau & Lifshitz 2003), this leads to a strong nonadiabatic coupling between the Edegenerate vibrational modes of the ion and the pwave states of the incident electron. The coupling is responsible for the fast DR rate (Kokoouline et al. 2001) in H_{3}^{+}. In the present model, only the pwave electronic states are included because other partial waves have a much smaller effect on the DR probability: the swave states have no Etype character and, therefore, are only weakly coupled to the dissociative electronic states of H_{3}; dwave electronic states are coupled to the Evibrational modes, but the coupling is rather small because the dwave of the incident electron does not penetrate sufficiently close to the ionic core owing to the dwave centrifugal potential barrier.
All three internal vibrational coordinates are taken into account. Vibrational dynamics of the ionic core are described using the hyperspherical coordinates, which represent the three vibrational degrees of freedom by a hyperradius and two hyperangles. The hyperradius is treated as a dissociation coordinate that represents uniformly the two possible DR channels, the threebody (such as H+H+H) and twobody (such as H_{2}+H). Although the initial vibrational state of the ion is the ground state, after recombination with the electron, other vibrational states of the ionic target molecule can be populated. Therefore, in general, many vibrational states have to be included in the treatment. In particular, the states of the vibrational continuum have to be included, because only such states can lead to the dissociation of the neutral molecule. The vibrational states of the continuum are obtained using a complex absorbing potential placed at a large hyperradius to absorb the flux of the outgoing dissociative wave.
Since the rovibrational symmetry is D_{3h} for H_{3}^{+} and D_{3}^{+} and C_{2v} for H_{2}D^{+} and D_{2}H^{+}, the rovibrational functions are classified according to the irreducible representations of the corresponding symmetry groups, i.e. A_{1}', A_{1}'', A_{2}', A_{2}'', E', and E'' for D_{3h} and A_{1}, A_{2}, B_{1}, and B_{2} for C_{2v}. We use the rigid rotor approximation, i.e. the vibrational and rotational parts of the total wave function are calculated independently by diagonalizing the corresponding Hamiltonians. In our approach, the rotational wave functions must be obtained separately for the ions and for the neutral molecules. They are constructed in a different way for the D_{3h} and C_{2v} cases. The rotational eigenstates and eigenenergies of the D_{3h} molecules are symmetric top wave functions (see, for example Bunker & Jensen 1998). They can be obtained analytically if the rotational constants are known. The rotational constants are obtained numerically from vibrational wave functions, i.e. they are calculated separately for each vibrational level of the target molecule. The rotational functions for the C_{2v} ions are obtained numerically by diagonalizing the asymmetric top Hamiltonian (Kokoouline & Greene 2005; Bunker & Jensen 1998).
Once the rovibrational wave functions are calculated, we construct the electronion scattering matrix (Smatrix). The Smatrix is calculated in the framework of quantum defect theory (QDT) (see, for example, Chang & Fano 1972; Kokoouline & Greene 2005,2003b) using the quantum defect parameters obtained from the ab initio calculation (Mistrik et al. 2001). The constructed scattering matrix accounts for the JahnTeller effect and is diagonal with respect to the different irreducible representations and the total angular momentum N of the neutral molecule. Thus, the actual calculations are made separately for each and N. Elements of the matrix describe the scattering amplitudes for the change of the rovibrational state of the ion after a collision with the electron. However, the Smatrix is not unitary due to the presence of the dissociative vibrational channels (i.e. continuum vibrational states of the ion, discussed above), which are not explicitly listed in the computed Smatrix. The ``defect'' from unitarity of each column of this Smatrix is associated with the dissociation probability of the neutral molecule formed during the scattering process. The dissociation probability per collision is then used to calculate the DR crosssections and rate coefficients.
The nuclear spin states are characterized by one of the A_{1}, A_{2}, or E irreducible representations of the symmetry group S_{3} for D_{3h} molecules and by the A or B irreducible representations of the symmetry group S_{2} for C_{2v} molecules. The irreducible representation of a particular nuclear spin state determines its statistical weight and is related to the total nuclear spin of the state. Here, is the vector sum of spins of identical nuclei.
For H_{3}^{+}, the states (A_{2}' and A_{2}'' rovibrational states) correspond to I=3/2 (ortho); the states (E' and E'' rovibrational states) correspond to I=1/2 (para). The statistical ortho:para weights are 2:1.
For H_{2}D^{+}, the states (B_{1} and B_{2} rovibrational states) correspond to I=1 (ortho); the states (A_{1} and A_{2} rovibrational states) correspond to I=0 (para). The statistical ortho:para weights are 3:1.
For D_{2}H^{+}, the states (A_{1} and A_{2} rovibrational states) correspond to I=0,2 (ortho); the states (B_{1} and B_{2} rovibrational states) correspond to I=1 (para). The statistical ortho:para weights are 2:1.
Finally, for D_{3}^{+}, the states (A_{1}' and A_{1}'' rovibrational states) correspond to I=1,3 (ortho); the states (A_{2}' and A_{2}'' rovibrational states) correspond to I=0 (para); the states (E' and E'' rovibrational states) correspond to I=1,2 (meta). The statistical ortho:para:meta weights are 10:1:8.
Figure B.1: Theoretical DR rate coefficients as functions of temperature for the ortho and paraspecies of H_{3}^{+}. The figure also shows the speciesaveraged rate coefficient. For comparison, we show the rate coefficient obtained from measurements in the TSR storage ring by Kreckel et al. (2005) and the analytical dependence for the coefficient used in earlier models of prestellar cores by FPdFW. 

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Figure B.2: Theoretical DR rate coefficients for the ortho and paraspecies of H_{2}D^{+}, as functions of temperature. The rate coefficient averaged over the two species and the analytical expression used in earlier models are also shown. 

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Figure B.3: Theoretical DR rate coefficients for the ortho and paraspecies of D_{2}H^{+} as a function of temperature. The rate coefficient averaged over the two species and the analytical expression used in earlier models are also shown. 

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Figure B.4: Theoretical DR rate coefficients for the ortho, para, and metaspecies of D_{3}^{+}. The rate coefficient averaged over the two species and the one used in earlier models are also shown. 

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Figures (B.1)(B.4) summarize the obtained DR thermal rate coefficients calculated separately for each nuclear spin species of the four H_{3}^{+} isotopologues and the numerical values are listed in Table B.1. For comparison, the figures also show the analytical dependences used in previous models of prestellar core chemistry (FPdFW). As one can see, the rates for different nuclear spin species are similar to each other (for a given isotopologue) at high temperatures. However, for lower temperatures, the rates for different ortho/para/metanuclear spin species significantly differ from each other. The difference in behavior at low temperatures is explained by different energies of Rydberg resonances present in DR crosssections at low electron energies. The actual energies of such resonances are important for the thermal average at temperatures below or similar to the energy difference between ground rotational levels of different nuclear spin species. At higher temperatures, the exact energy of the resonances is not important. The averaged rate is determined by the density and the widths of the resonances, which are similar for all nuclear spin species over a large range of collision energies.
Table B.1: Dissociative recombination rates of H_{3}^{+}, H_{2}D^{+}, D_{2}H^{+}, and D_{3}^{+} for each individual nuclear spin state species.
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